Problem 20 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) is inversely proportional to \(x,\) and \(y=21\) when \(x=3\).

Step-by-Step Solution

Answer

The model is \( y = \frac{63}{x} \).

Step 1: Understand Inverse Proportionality

When a variable is inversely proportional to another variable, this relationship can be expressed as \( y = \frac{k}{x} \), where \( k \) is a constant. This means as \( x \) increases, \( y \) decreases and vice versa.

Step 2: Identify Given Values

We are informed that \( y = 21 \) when \( x = 3 \). We need to use these values to find the constant \( k \).

Step 3: Solve for the Constant \( k \)

Substitute the given values \( y = 21 \) and \( x = 3 \) into the equation \( y = \frac{k}{x} \):\[21 = \frac{k}{3}\]Multiply both sides by 3 to isolate \( k \):\[k = 21 \times 3 = 63\]

Step 4: Construct the Model

Using the constant \( k = 63 \), substitute back into the general inverse proportionality model:\[ y = \frac{63}{x} \]This is the mathematical model that represents the situation given.

Key Concepts

Constant of Proportionality

In any inverse proportionality relationship, the concept of a constant of proportionality is crucial. This constant, often denoted as \( k \), plays a vital role in preserving the relationship between two inversely related variables. In our specific example, as \( y \) is inversely proportional to \( x \), we use the equation \( y = \frac{k}{x} \). Here, \( k \) is our constant of proportionality.
To determine \( k \), we insert the values given in the problem: \( y = 21 \) and \( x = 3 \). By substituting, the equation becomes:\[21 = \frac{k}{3}\]By solving this, we multiply both sides by 3, revealing that \( k = 63 \).
In essence, the constant of proportionality \( k \) ensures that no matter the values of \( x \) and \( y \), the ratio respective to their relationship stays consistent as dictated by \( k \). This constancy allows us to predict or model new scenarios under the same conditions.

Mathematical Models

Creating a mathematical model like we did here allows us to represent real-world relationships in a structured mathematical format. When we say that \( y \) is inversely proportional to \( x \), we craft a model using the formula \( y = \frac{k}{x} \).
This model now tells us how two variables interact with each other in an ideal inverse setting. By determining the constant \( k \) through the given conditions, which was 63 in our example, we established a complete model: \[ y = \frac{63}{x} \].
Such models are powerful as they give us the ability to predict how changes in one variable might impact the other. In practical terms, if you know \( x \) and you apply the model, you can directly solve for \( y \), and vice versa. Therefore, mathematical modeling converts complex real-world interactions into something computable and understandable.

Understanding Relationships in Algebra

Understanding how variables relate within equations is a fundamental aspect of algebra. The concept of inverse proportionality is a great illustration of this. The relationship described by \( y = \frac{k}{x} \) gives insight into how \( y \) and \( x \) move in opposite directions. If \( x \) becomes larger, \( y \) will decrease, and the opposite is true as well. This interaction is part of what makes algebra powerful.
By understanding these relationships, you are not just solving an equation; you are peering into how quantities relate and affect each other, which is applicable in diverse fields such as physics, economics, and biology. Algebra enables you to construct relationships that can be used to solve real-world problems, predict outcomes, and understand patterns.
As you explore more algebraic relationships, keep in mind that they often reveal deeper insights about the world around us. Each equation, like the inverse proportionality, tells a unique story about how variables influence each other.